CINXE.COM

<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.1 plus MathML 2.0 plus SVG 1.1//EN" "http://www.w3.org/2002/04/xhtml-math-svg/xhtml-math-svg-flat.dtd" > <html xmlns="http://www.w3.org/1999/xhtml"> <head> <title> looping in nLab </title> <meta http-equiv="Content-Type" content="text/html; charset=UTF-8" /> <meta name="robots" content="index,follow" /> <meta name="viewport" content="width=device-width, initial-scale=1" /> <link href="/stylesheets/instiki.css?1676280126" media="all" rel="stylesheet" type="text/css" /> <link href="/stylesheets/mathematics.css?1660229990" media="all" rel="stylesheet" type="text/css" /> <link href="/stylesheets/syntax.css?1660229990" media="all" rel="stylesheet" type="text/css" /> <link href="/stylesheets/nlab.css?1676280126" media="all" rel="stylesheet" type="text/css" /> <link rel="stylesheet" type="text/css" href="https://cdn.jsdelivr.net/gh/dreampulse/computer-modern-web-font@master/fonts.css"/> <style type="text/css"> h1#pageName, div.info, .newWikiWord a, a.existingWikiWord, .newWikiWord a:hover, [actiontype="toggle"]:hover, #TextileHelp h3 { color: #226622; } a:visited.existingWikiWord { color: #164416; } </style> <style type="text/css"><![CDATA[/*></style> <script src="/javascripts/prototype.js?1660229990" type="text/javascript"></script> <script src="/javascripts/effects.js?1660229990" type="text/javascript"></script> <script src="/javascripts/dragdrop.js?1660229990" type="text/javascript"></script> <script src="/javascripts/controls.js?1660229990" type="text/javascript"></script> <script src="/javascripts/application.js?1660229990" type="text/javascript"></script> <script src="/javascripts/page_helper.js?1660229990" type="text/javascript"></script> <script src="/javascripts/thm_numbering.js?1660229990" type="text/javascript"></script> <script type="text/x-mathjax-config"> <![CDATA[//><!]]> </script> <script type="text/javascript"> <![CDATA[//><!]]> </script> <link href="https://ncatlab.org/nlab/atom_with_headlines" rel="alternate" title="Atom with headlines" type="application/atom+xml" /> <link href="https://ncatlab.org/nlab/atom_with_content" rel="alternate" title="Atom with full content" type="application/atom+xml" /> <script type="text/javascript"> document.observe("dom:loaded", function() { generateThmNumbers(); }); </script> </head> <body> <div id="Container"> <div id="Content"> <h1 id="pageName"> <span style="float: left; margin: 0.5em 0.25em -0.25em 0"> <svg xmlns="http://www.w3.org/2000/svg" width="1.872em" height="1.8em" viewBox="0 0 190 181"> <path fill="#226622" d="M72.8 145c-1.6 17.3-15.7 10-23.6 20.2-5.6 7.3 4.8 15 11.4 15 11.5-.2 19-13.4 26.4-20.3 3.3-3 8.2-4 11.2-7.2a14 14 0 0 0 2.9-11.1c-1.4-9.6-12.4-18.6-16.9-27.2-5-9.6-10.7-27.4-24.1-27.7-17.4-.3-.4 26 4.7 30.7 2.4 2.3 5.4 4.1 7.3 6.9 1.6 2.3 2.1 5.8-1 7.2-5.9 2.6-12.4-6.3-15.5-10-8.8-10.6-15.5-23-26.2-31.8-5.2-4.3-11.8-8-18-3.7-7.3 4.9-4.2 12.9.2 18.5a81 81 0 0 0 30.7 23c3.3 1.5 12.8 5.6 10 10.7-2.5 5.2-11.7 3-15.6 1.1-8.4-3.8-24.3-21.3-34.4-13.7-3.5 2.6-2.3 7.6-1.2 11.1 2.8 9 12.2 17.2 20.9 20.5 17.3 6.7 34.3-8 50.8-12.1z"/> <path fill="#a41e32" d="M145.9 121.3c-.2-7.5 0-19.6-4.5-26-5.4-7.5-12.9-1-14.1 5.8-1.4 7.8 2.7 14.1 4.8 21.3 3.4 12 5.8 29-.8 40.1-3.6-6.7-5.2-13-7-20.4-2.1-8.2-12.8-13.2-15.1-1.9-2 9.7 9 21.2 12 30.1 1.2 4 2 8.8 6.4 10.3 6.9 2.3 13.3-4.7 17.7-8.8 12.2-11.5 36.6-20.7 43.4-36.4 6.7-15.7-13.7-14-21.3-7.2-9.1 8-11.9 20.5-23.6 25.1 7.5-23.7 31.8-37.6 38.4-61.4 2-7.3-.8-29.6-13-19.8-14.5 11.6-6.6 37.6-23.3 49.2z"/> <path fill="#193c78" d="M86.3 47.5c0-13-10.2-27.6-5.8-40.4 2.8-8.4 14.1-10.1 17-1 3.8 11.6-.3 26.3-1.8 38 11.7-.7 10.5-16 14.8-24.3 2.1-4.2 5.7-9.1 11-6.7 6 2.7 7.4 9.2 6.6 15.1-2.2 14-12.2 18.8-22.4 27-3.4 2.7-8 6.6-5.9 11.6 2 4.4 7 4.5 10.7 2.8 7.4-3.3 13.4-16.5 21.7-16 14.6.7 12 21.9.9 26.2-5 1.9-10.2 2.3-15.2 3.9-5.8 1.8-9.4 8.7-15.7 8.9-6.1.1-9-6.9-14.3-9-14.4-6-33.3-2-44.7-14.7-3.7-4.2-9.6-12-4.9-17.4 9.3-10.7 28 7.2 35.7 12 2 1.1 11 6.9 11.4 1.1.4-5.2-10-8.2-13.5-10-11.1-5.2-30-15.3-35-27.3-2.5-6 2.8-13.8 9.4-13.6 6.9.2 13.4 7 17.5 12C70.9 34 75 43.8 86.3 47.4z"/> </svg> </span> <span class="webName">nLab</span> looping </h1> <div class="navigation"> <span class="skipNav"><a href='#navEnd'>Skip the Navigation Links</a> | </span> <span style="display:inline-block; width: 0.3em;"></span> <a href="/nlab/show/HomePage" accesskey="H" title="Home page">Home Page</a> | <a href="/nlab/all_pages" accesskey="A" title="List of all pages">All Pages</a> | <a href="/nlab/latest_revisions" accesskey="U" title="Latest edits and page creations">Latest Revisions</a> | <a href="https://nforum.ncatlab.org/discussion/11586/#Item_2" title="Discuss this page in its dedicated thread on the nForum" style="color: black">Discuss this page</a> | <form accept-charset="utf-8" action="/nlab/search" id="navigationSearchForm" method="get"> <fieldset class="search"><input type="text" id="searchField" name="query" value="Search" style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="higher_category_theory">Higher category theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/higher+category+theory">higher category theory</a></strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a></li> <li><a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a></li> </ul> <h2 id="basic_concepts">Basic concepts</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/k-morphism">k-morphism</a>, <a class="existingWikiWord" href="/nlab/show/coherence">coherence</a></li> <li><a class="existingWikiWord" href="/nlab/show/looping+and+delooping">looping and delooping</a></li> <li><a class="existingWikiWord" href="/nlab/show/stabilization">looping and suspension</a></li> </ul> <h2 id="basic_theorems">Basic theorems</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+hypothesis">homotopy hypothesis</a>-theorem</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/delooping+hypothesis">delooping hypothesis</a>-theorem</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/periodic+table">periodic table</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/stabilization+hypothesis">stabilization hypothesis</a>-theorem</p> </li> <li> <p><a class="existingWikiWord" href="/michaelshulman/show/exactness+hypothesis">exactness hypothesis</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/holographic+principle+of+higher+category+theory">holographic principle</a></p> </li> </ul> <h2 id="applications">Applications</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/applications+of+%28higher%29+category+theory">applications of (higher) category theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+category+theory+and+physics">higher category theory and physics</a></p> </li> </ul> <h2 id="models">Models</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/%28n%2Cr%29-category">(n,r)-category</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/Theta-space">Theta-space</a></li> <li><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-category">∞-category</a>/<a class="existingWikiWord" href="/nlab/show/%E2%88%9E-category">∞-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2Cn%29-category">(∞,n)-category</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/n-fold+complete+Segal+space">n-fold complete Segal space</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C2%29-category">(∞,2)-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/quasi-category">quasi-category</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/algebraic+quasi-category">algebraic quasi-category</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/simplicially+enriched+category">simplicially enriched category</a></li> <li><a class="existingWikiWord" href="/nlab/show/complete+Segal+space">complete Segal space</a></li> <li><a class="existingWikiWord" href="/nlab/show/model+category">model category</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C0%29-category">(∞,0)-category</a>/<a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groupoid">∞-groupoid</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/Kan+complex">Kan complex</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/algebraic+Kan+complex">algebraic Kan complex</a></li> <li><a class="existingWikiWord" href="/nlab/show/simplicial+T-complex">simplicial T-complex</a></li> </ul> </li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2CZ%29-category">(∞,Z)-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/n-category">n-category</a> = (n,n)-category <ul> <li><a class="existingWikiWord" href="/nlab/show/2-category">2-category</a>, <a class="existingWikiWord" href="/nlab/show/%282%2C1%29-category">(2,1)-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/1-category">1-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/0-category">0-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/%28-1%29-category">(-1)-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/%28-2%29-category">(-2)-category</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/n-poset">n-poset</a> = <a class="existingWikiWord" href="/nlab/show/n-poset">(n-1,n)-category</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/poset">poset</a> = <a class="existingWikiWord" href="/nlab/show/%280%2C1%29-category">(0,1)-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/2-poset">2-poset</a> = <a class="existingWikiWord" href="/nlab/show/%281%2C2%29-category">(1,2)-category</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/n-groupoid">n-groupoid</a> = (n,0)-category <ul> <li><a class="existingWikiWord" href="/nlab/show/2-groupoid">2-groupoid</a>, <a class="existingWikiWord" href="/nlab/show/3-groupoid">3-groupoid</a></li> </ul> </li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/categorification">categorification</a>/<a class="existingWikiWord" href="/nlab/show/decategorification">decategorification</a></li> <li><a class="existingWikiWord" href="/nlab/show/geometric+definition+of+higher+category">geometric definition of higher category</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/Kan+complex">Kan complex</a></li> <li><a class="existingWikiWord" href="/nlab/show/quasi-category">quasi-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/simplicial+model+for+weak+%E2%88%9E-categories">simplicial model for weak ∞-categories</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/complicial+set">complicial set</a></li> <li><a class="existingWikiWord" href="/nlab/show/weak+complicial+set">weak complicial set</a></li> </ul> </li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/algebraic+definition+of+higher+category">algebraic definition of higher category</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/bicategory">bicategory</a></li> <li><a class="existingWikiWord" href="/nlab/show/bigroupoid">bigroupoid</a></li> <li><a class="existingWikiWord" href="/nlab/show/tricategory">tricategory</a></li> <li><a class="existingWikiWord" href="/nlab/show/tetracategory">tetracategory</a></li> <li><a class="existingWikiWord" href="/nlab/show/strict+%E2%88%9E-category">strict ∞-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/Batanin+%E2%88%9E-category">Batanin ∞-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/Trimble+n-category">Trimble ∞-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/Grothendieck-Maltsiniotis+%E2%88%9E-categories">Grothendieck-Maltsiniotis ∞-categories</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/stable+homotopy+theory">stable homotopy theory</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric monoidal category</a></li> <li><a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+%28%E2%88%9E%2C1%29-category">symmetric monoidal (∞,1)-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/stable+%28%E2%88%9E%2C1%29-category">stable (∞,1)-category</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/dg-category">dg-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E+category">A-∞ category</a></li> <li><a class="existingWikiWord" href="/nlab/show/triangulated+category">triangulated category</a></li> </ul> </li> </ul> </li> </ul> <h2 id="morphisms">Morphisms</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/k-morphism">k-morphism</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/2-morphism">2-morphism</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/transfor">transfor</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/natural+transformation">natural transformation</a></li> <li><a class="existingWikiWord" href="/nlab/show/modification">modification</a></li> </ul> </li> </ul> <h2 id="functors">Functors</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/functor">functor</a></li> <li><a class="existingWikiWord" href="/nlab/show/2-functor">2-functor</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/pseudofunctor">pseudofunctor</a></li> <li><a class="existingWikiWord" href="/nlab/show/lax+functor">lax functor</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-functor">(∞,1)-functor</a></li> </ul> <h2 id="universal_constructions">Universal constructions</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/2-limit">2-limit</a></li> <li><a class="existingWikiWord" href="/nlab/show/adjoint+%28%E2%88%9E%2C1%29-functor">(∞,1)-adjunction</a></li> <li><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-Kan+extension">(∞,1)-Kan extension</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/limit+in+a+quasi-category">(∞,1)-limit</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-Grothendieck+construction">(∞,1)-Grothendieck construction</a></li> </ul> <h2 id="extra_properties_and_structure">Extra properties and structure</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/cosmic+cube">cosmic cube</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/k-tuply+monoidal+n-category">k-tuply monoidal n-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/strict+%E2%88%9E-category">strict ∞-category</a>, <a class="existingWikiWord" href="/nlab/show/strict+%E2%88%9E-groupoid">strict ∞-groupoid</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/stable+%28%E2%88%9E%2C1%29-category">stable (∞,1)-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a></li> </ul> <h2 id="1categorical_presentations">1-categorical presentations</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/homotopical+category">homotopical category</a></li> <li><a class="existingWikiWord" href="/nlab/show/model+category">model category theory</a></li> <li><a class="existingWikiWord" href="/nlab/show/enriched+category+theory">enriched category theory</a></li> </ul> </div></div> <h4 id="homotopy_theory">Homotopy theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a>, <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+theory">(∞,1)-category theory</a>, <a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theory</a></strong></p> <p>flavors: <a class="existingWikiWord" href="/nlab/show/stable+homotopy+theory">stable</a>, <a class="existingWikiWord" href="/nlab/show/equivariant+homotopy+theory">equivariant</a>, <a class="existingWikiWord" href="/nlab/show/rational+homotopy+theory">rational</a>, <a class="existingWikiWord" href="/nlab/show/p-adic+homotopy+theory">p-adic</a>, <a class="existingWikiWord" href="/nlab/show/proper+homotopy+theory">proper</a>, <a class="existingWikiWord" href="/nlab/show/geometric+homotopy+theory">geometric</a>, <a class="existingWikiWord" href="/nlab/show/cohesive+homotopy+theory">cohesive</a>, <a class="existingWikiWord" href="/nlab/show/directed+homotopy+theory">directed</a>…</p> <p>models: <a class="existingWikiWord" href="/nlab/show/topological+homotopy+theory">topological</a>, <a class="existingWikiWord" href="/nlab/show/simplicial+homotopy+theory">simplicial</a>, <a class="existingWikiWord" href="/nlab/show/localic+homotopy+theory">localic</a>, …</p> <p>see also <strong><a class="existingWikiWord" href="/nlab/show/algebraic+topology">algebraic topology</a></strong></p> <p><strong>Introductions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Introduction+to+Topology+--+2">Introduction to Basic Homotopy Theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Introduction+to+Homotopy+Theory">Introduction to Abstract Homotopy Theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+homotopy+types">geometry of physics – homotopy types</a></p> </li> </ul> <p><strong>Definitions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy">homotopy</a>, <a class="existingWikiWord" href="/nlab/show/higher+homotopy">higher homotopy</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+type">homotopy type</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Pi-algebra">Pi-algebra</a>, <a class="existingWikiWord" href="/nlab/show/spherical+object+and+Pi%28A%29-algebra">spherical object and Pi(A)-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+coherent+category+theory">homotopy coherent category theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopical+category">homotopical category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+category">model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/category+of+fibrant+objects">category of fibrant objects</a>, <a class="existingWikiWord" href="/nlab/show/cofibration+category">cofibration category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Waldhausen+category">Waldhausen category</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+category">homotopy category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Ho%28Top%29">Ho(Top)</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/homotopy+category+of+an+%28%E2%88%9E%2C1%29-category">homotopy category of an (∞,1)-category</a></li> </ul> </li> </ul> <p><strong>Paths and cylinders</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/left+homotopy">left homotopy</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cylinder+object">cylinder object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+cone">mapping cone</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/right+homotopy">right homotopy</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/path+object">path object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+cocone">mapping cocone</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/generalized+universal+bundle">universal bundle</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/interval+object">interval object</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+localization">homotopy localization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/infinitesimal+interval+object">infinitesimal interval object</a></p> </li> </ul> </li> </ul> <p><strong>Homotopy groups</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+group">homotopy group</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+group">fundamental group</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/fundamental+group+of+a+topos">fundamental group of a topos</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Brown-Grossman+homotopy+group">Brown-Grossman homotopy group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/categorical+homotopy+groups+in+an+%28%E2%88%9E%2C1%29-topos">categorical homotopy groups in an (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+homotopy+groups+in+an+%28%E2%88%9E%2C1%29-topos">geometric homotopy groups in an (∞,1)-topos</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid">fundamental ∞-groupoid</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+groupoid">fundamental groupoid</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/path+groupoid">path groupoid</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid+in+a+locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid+of+a+locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">fundamental ∞-groupoid of a locally ∞-connected (∞,1)-topos</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%28%E2%88%9E%2C1%29-category">fundamental (∞,1)-category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/fundamental+category">fundamental category</a></li> </ul> </li> </ul> <p><strong>Basic facts</strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/fundamental+group+of+the+circle+is+the+integers">fundamental group of the circle is the integers</a></li> </ul> <p><strong>Theorems</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+theorem+of+covering+spaces">fundamental theorem of covering spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Freudenthal+suspension+theorem">Freudenthal suspension theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Blakers-Massey+theorem">Blakers-Massey theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+homotopy+van+Kampen+theorem">higher homotopy van Kampen theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/nerve+theorem">nerve theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Whitehead%27s+theorem">Whitehead's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hurewicz+theorem">Hurewicz theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Galois+theory">Galois theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+hypothesis">homotopy hypothesis</a>-theorem</p> </li> </ul> </div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#Examples'>Examples</a></li> <ul> <li><a href='#ForPlainGroupsDeloopingToGroupoids'>For plain groups delooping to groupoids</a></li> <li><a href='#for_topological_spaces_and_groupoids'>For topological spaces and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-groupoids</a></li> <li><a href='#for_parameterized_groupoids_stacks__sheaves'>For parameterized <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-groupoids (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-stacks / <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-sheaves)</a></li> <li><a href='#for_cohesive_groupoids'>For cohesive <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-groupoids</a></li> <li><a href='#for_categories'>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,n)</annotation></semantics></math>-categories</a></li> </ul> <li><a href='#relation_to_looping_and_suspension'>Relation to looping and suspension</a></li> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> any kind of <a class="existingWikiWord" href="/nlab/show/space">space</a> (or possibly a <a class="existingWikiWord" href="/nlab/show/directed+space">directed space</a>, viewed as some sort of <a class="existingWikiWord" href="/nlab/show/category">category</a> or higher dimensional analogue of one), its <a class="existingWikiWord" href="/nlab/show/loop+space+object">loop space object</a>s <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Ω</mi> <mi>x</mi></msub><mi>X</mi></mrow><annotation encoding="application/x-tex">\Omega_x X</annotation></semantics></math> canonically inherit a <a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal structure</a>, coming from concatenation of <a class="existingWikiWord" href="/nlab/show/loops">loops</a>.</p> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">x \in X</annotation></semantics></math> is essentially unique, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Ω</mi> <mi>x</mi></msub><mi>X</mi></mrow><annotation encoding="application/x-tex">\Omega_x X</annotation></semantics></math> equipped with this monoidal structure remembers all of the structure of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>: we say <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>≃</mo><mi>B</mi><msub><mi>Ω</mi> <mi>x</mi></msub><mi>X</mi></mrow><annotation encoding="application/x-tex">X \simeq B \Omega_x X</annotation></semantics></math> and call <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mi>A</mi></mrow><annotation encoding="application/x-tex">B A</annotation></semantics></math> the <em><a class="existingWikiWord" href="/nlab/show/delooping">delooping</a></em> of the monoidal object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>.</p> <p>What all these terms (“loops” <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ω</mi></mrow><annotation encoding="application/x-tex">\Omega</annotation></semantics></math>, “delooping” <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math> etc.) mean in detail and how they are <em>presented</em> concretely depends on the given setup. We discuss some of these below in the section <a href="#Examples">Examples</a>.</p> <h2 id="Examples">Examples</h2> <h3 id="ForPlainGroupsDeloopingToGroupoids">For plain groups delooping to groupoids</h3> <p>Write</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Grpd">Grpd</a> for the <a class="existingWikiWord" href="/nlab/show/%282%2C1%29-category">(2,1)-category</a> of <a class="existingWikiWord" href="/nlab/show/groupoids">groupoids</a> (<a class="existingWikiWord" href="/nlab/show/objects">objects</a> are <a class="existingWikiWord" href="/nlab/show/groupoids">groupoids</a>, <a class="existingWikiWord" href="/nlab/show/1-morphisms">1-morphisms</a> are <a class="existingWikiWord" href="/nlab/show/functors">functors</a> between these and <a class="existingWikiWord" href="/nlab/show/2-morphisms">2-morphisms</a> are <a class="existingWikiWord" href="/nlab/show/natural+transformations">natural transformations</a> between those, which are necessarily <a class="existingWikiWord" href="/nlab/show/natural+isomorphisms">natural isomorphisms</a>),</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Grp">Grp</a> for the <a class="existingWikiWord" href="/nlab/show/1-category">1-category</a> of <a class="existingWikiWord" href="/nlab/show/groups">groups</a> (<a class="existingWikiWord" href="/nlab/show/discrete+groups">discrete groups</a>), also regarded as a <a class="existingWikiWord" href="/nlab/show/%282%2C1%29-category">(2,1)-category</a>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Grpd</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msup></mrow><annotation encoding="application/x-tex">Grpd^{\ast/}</annotation></semantics></math> for the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>2</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(2,1)</annotation></semantics></math>-category of <a class="existingWikiWord" href="/nlab/show/pointed+objects">pointed objects</a> in <a class="existingWikiWord" href="/nlab/show/Grpd">Grpd</a>,</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Grpd</mi> <mrow><mo>≥</mo><mn>1</mn></mrow></msub><mo>↪</mo><mi>Grpd</mi></mrow><annotation encoding="application/x-tex">Grpd_{\geq 1} \hookrightarrow Grpd</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/full+sub-%28infinity%2C1%29-category">full sub-(2,1)-category</a> of <a class="existingWikiWord" href="/nlab/show/connected+object+in+an+%28infinity%2C1%29-topos">connected</a> groupoids, those for which <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>0</mn></msub><mo>≃</mo><mo>*</mo></mrow><annotation encoding="application/x-tex">\pi_0 \simeq \ast</annotation></semantics></math>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>Grp</mi> <mrow><mo>≥</mo><mn>1</mn></mrow> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msubsup></mrow><annotation encoding="application/x-tex">Grp^{\ast/}_{\geq 1}</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/pointed+objects">pointed objects</a> in connected groupoids.</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo><mo>∈</mo><mi>Grp</mi></mrow><annotation encoding="application/x-tex">\pi_1(X,x) \in Grp</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/fundamental+group">fundamental group</a> of a pointed groupoid <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo>*</mo><mover><mo>→</mo><mi>x</mi></mover><mi>X</mi><mo stretchy="false">)</mo><mo>∈</mo><msup><mi>Grpd</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msup></mrow><annotation encoding="application/x-tex">(\ast \stackrel{x}{\to} X) \in Grpd^{\ast/}</annotation></semantics></math> at the given basepoint.</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo>∈</mo><mi>Grpd</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}G \in Grpd</annotation></semantics></math>, given a group <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>, for the groupoid <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>G</mi><mover><mo>⟶</mo><mo>⟶</mo></mover><mo>*</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(G\stackrel{\longrightarrow}{\longrightarrow} \ast)</annotation></semantics></math>, with composition given by the product in the group. There are two possible choices of conventions, we agree that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mo>*</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>g</mi> <mn>1</mn></msub></mrow></mpadded></msup><mo>↗</mo></mtd> <mtd></mtd> <mtd><msup><mo>↘</mo> <mpadded width="0"><mrow><msub><mi>g</mi> <mn>2</mn></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mo>*</mo></mtd> <mtd></mtd> <mtd><munder><mo>⟶</mo><mrow><msub><mi>g</mi> <mn>1</mn></msub><mo>⋅</mo><msub><mi>g</mi> <mn>2</mn></msub></mrow></munder></mtd> <mtd></mtd> <mtd><mo>*</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ && \ast \\ & {}^{\mathllap{g_1}}\nearrow && \searrow^{\mathrlap{g_2}} \\ \ast && \underset{g_1 \cdot g_2}{\longrightarrow} && \ast } \,. </annotation></semantics></math></div></li> </ul> <div class="num_prop" id="SkeletalRepresentativesForConnectedGroupoids"> <h6 id="proposition">Proposition</h6> <p>The <a class="existingWikiWord" href="/nlab/show/%282%2C1%29-category">(2,1)-category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Grp</mi> <mrow><mo>≥</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">Grp_{\geq 1}</annotation></semantics></math> of connected groupoids is equivalent to its <a class="existingWikiWord" href="/nlab/show/full+sub-%28infinity%2C1%29-category">full sub-(2,1)-category</a> on those objects of the form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}G</annotation></semantics></math>, for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> a group.</p> </div> <div class="proof"> <h6 id="proof">Proof</h6> <p>Given a connected groupoid <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>, pick any basepoint <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">x\in X</annotation></semantics></math> and consider the canonical inclusion <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo><mo>⟶</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}\pi_1(X,x) \longrightarrow X</annotation></semantics></math>. By construction this is <a class="existingWikiWord" href="/nlab/show/fully+faithful+functor">fully faithful</a> and by assumption of connectedness it is <a class="existingWikiWord" href="/nlab/show/essentially+surjective+functor">essentially surjective</a>, hence it is an <a class="existingWikiWord" href="/nlab/show/equivalence+of+groupoids">equivalence of groupoids</a>.</p> </div> <div class="num_prop" id="HomsBetweenBGs"> <h6 id="proposition_2">Proposition</h6> <p>The <a class="existingWikiWord" href="/nlab/show/hom-groupoids">hom-groupoids</a> between connected groupoids with fundamental groups <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi></mrow><annotation encoding="application/x-tex">H</annotation></semantics></math>, respectively, are equivalent to the <a class="existingWikiWord" href="/nlab/show/action+groupoids">action groupoids</a> of the set of group <a class="existingWikiWord" href="/nlab/show/homomorphisms">homomorphisms</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo>→</mo><mi>H</mi></mrow><annotation encoding="application/x-tex">G \to H</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/action">acted</a> on by <a class="existingWikiWord" href="/nlab/show/conjugation">conjugation</a> with elements of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi></mrow><annotation encoding="application/x-tex">H</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Grpd</mi><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo>,</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>H</mi><mo stretchy="false">)</mo><mo>≃</mo><mi>Grp</mi><mo stretchy="false">(</mo><mi>G</mi><mo>,</mo><mi>H</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><msub><mo stretchy="false">/</mo> <mi>Ad</mi></msub><mi>H</mi></mrow><annotation encoding="application/x-tex"> Grpd(\mathbf{B}G, \mathbf{B}H) \simeq Grp(G,H)//_{Ad}H </annotation></semantics></math></div> <p>Given two group homomorphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ϕ</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>ϕ</mi> <mn>2</mn></msub><mo lspace="verythinmathspace">:</mo><mi>G</mi><mo>⟶</mo><mi>H</mi></mrow><annotation encoding="application/x-tex">\phi_1, \phi_2 \colon G \longrightarrow H</annotation></semantics></math> then an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a> between them in this hom-groupoid is an element <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi><mo>∈</mo><mi>H</mi></mrow><annotation encoding="application/x-tex">h \in H</annotation></semantics></math> such that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>ϕ</mi> <mn>2</mn></msub><mo>=</mo><msub><mi>Ad</mi> <mi>h</mi></msub><mo>∘</mo><msub><mi>ϕ</mi> <mn>1</mn></msub><mo>≔</mo><msup><mi>h</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo>⋅</mo><msub><mi>ϕ</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>⋅</mo><mi>h</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \phi_2 = Ad_h \circ \phi_1 \coloneqq h^{-1}\cdot \phi_1(-) \cdot h \,. </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof_2">Proof</h6> <p>By direct inspection of the naturality square for the <a class="existingWikiWord" href="/nlab/show/natural+transformations">natural transformations</a> which are the morphisms in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Grpd</mi><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo>,</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>H</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Grpd(\mathbf{B}G, \mathbf{B}H)</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mo>*</mo></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mo>*</mo></mtd> <mtd><mover><mo>⟶</mo><mi>h</mi></mover></mtd> <mtd><mo>*</mo></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mi>g</mi> <mn>1</mn></msub></mrow></mpadded></msup></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mi>ϕ</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><msub><mi>g</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mrow></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mi>ϕ</mi> <mn>2</mn></msub><mo stretchy="false">(</mo><msub><mi>g</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mo>*</mo></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mo>*</mo></mtd> <mtd><mover><mo>⟶</mo><mi>h</mi></mover></mtd> <mtd><mo>*</mo></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mi>g</mi> <mn>2</mn></msub></mrow></mpadded></msup></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mi>ϕ</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><msub><mi>g</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mi>ϕ</mi> <mn>2</mn></msub><mo stretchy="false">(</mo><msub><mi>g</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mo>*</mo></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mo>*</mo></mtd> <mtd><mover><mo>⟶</mo><mi>h</mi></mover></mtd> <mtd><mo>*</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ \ast && && \ast &\stackrel{h}{\longrightarrow}& \ast \\ \downarrow^{\mathrlap{g_1}} && && \downarrow^{\mathrlap{\phi_1(g_1)}} && \downarrow^{\mathrlap{\phi_2(g_1)}} \\ \ast && && \ast &\stackrel{h}{\longrightarrow}& \ast \\ \downarrow^{\mathrlap{g_2}} && && \downarrow^{\mathrlap{\phi_1(g_2)}} && \downarrow^{\mathrlap{\phi_2(g_2)}} \\ \ast && && \ast &\stackrel{h}{\longrightarrow}& \ast } \,. </annotation></semantics></math></div></div> <div class="num_remark" id="piAs2Functor"> <h6 id="remark">Remark</h6> <p>The operation of forming <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">\pi_1</annotation></semantics></math> is equivalently the operation of forming the <a class="existingWikiWord" href="/nlab/show/homotopy+fiber+product">homotopy fiber product</a> of the point inclusion with itself, and hence extends to a <a class="existingWikiWord" href="/nlab/show/%282%2C1%29-functor">(2,1)-functor</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>1</mn></msub><mo lspace="verythinmathspace">:</mo><msup><mi>Grpd</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msup><mo>⟶</mo><mi>Grp</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \pi_1 \colon Grpd^{\ast/} \longrightarrow Grp \,. </annotation></semantics></math></div></div> <div class="num_prop"> <h6 id="proposition_3">Proposition</h6> <p>Restricted to <a class="existingWikiWord" href="/nlab/show/connected+object+in+an+%28infinity%2C1%29-topos">connected groupoids</a> among the pointed groupoids, the functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>1</mn></msub><mo lspace="verythinmathspace">:</mo><msubsup><mi>Grpd</mi> <mrow><mo>≥</mo><mn>1</mn></mrow> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msubsup><mo>⟶</mo><mi>Grp</mi></mrow><annotation encoding="application/x-tex">\pi_1 \colon Grpd^{\ast/}_{\geq 1} \longrightarrow Grp</annotation></semantics></math> of remark <a class="maruku-ref" href="#piAs2Functor"></a> is an <a class="existingWikiWord" href="/nlab/show/equivalence+of+%282%2C1%29-categories">equivalence of (2,1)-categories</a>.</p> </div> <div class="proof"> <h6 id="proof_3">Proof</h6> <p>It is clear that the functor is essentially surjective: for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> any <a class="existingWikiWord" href="/nlab/show/group">group</a> then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo>,</mo><mo>*</mo><mo stretchy="false">)</mo><mo>≃</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">\pi_1(\mathbf{B}G,\ast) \simeq G</annotation></semantics></math>.</p> <p>The more interesting point to notice is that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">\pi_1</annotation></semantics></math> is indeed a fully faithful <a class="existingWikiWord" href="/nlab/show/%282%2C1%29-functor">(2,1)-functor</a>, in that for any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>∈</mo><msubsup><mi>Grpd</mi> <mrow><mo>≥</mo><mn>1</mn></mrow> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msubsup></mrow><annotation encoding="application/x-tex">(X,x), (Y,y) \in Grpd^{\ast/}_{\geq 1}</annotation></semantics></math> then the functor</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>π</mi> <mn>1</mn></msub><msub><mo stretchy="false">)</mo> <mrow><mi>X</mi><mo>,</mo><mi>Y</mi></mrow></msub><mo lspace="verythinmathspace">:</mo><msup><mi>Grpd</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msup><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>,</mo><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>⟶</mo><mi>Grp</mi><mo stretchy="false">(</mo><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> (\pi_1)_{X,Y} \colon Grpd^{\ast/}((X,y),(Y,y)) \longrightarrow Grp(\pi_1(X,x), \pi_1(Y,y)) </annotation></semantics></math></div> <p>is an <a class="existingWikiWord" href="/nlab/show/equivalence+of+groupoids">equivalence</a> of <a class="existingWikiWord" href="/nlab/show/hom-groupoids">hom-groupoids</a>. By prop. <a class="maruku-ref" href="#SkeletalRepresentativesForConnectedGroupoids"></a> it is sufficient to check this for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>=</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">X = \mathbf{B}G</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi><mo>=</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>H</mi></mrow><annotation encoding="application/x-tex">Y = \mathbf{B}H</annotation></semantics></math> with their canonical basepoints, hence to check that for any two groups <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo>,</mo><mi>H</mi></mrow><annotation encoding="application/x-tex">G,H</annotation></semantics></math> the functor</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>π</mi> <mn>1</mn></msub><msub><mo stretchy="false">)</mo> <mrow><mi>X</mi><mo>,</mo><mi>Y</mi></mrow></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msup><mi>Grpd</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msup><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo>,</mo><mo>*</mo><mo stretchy="false">)</mo><mo>,</mo><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>H</mi><mo>,</mo><mo>*</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>⟶</mo><mi>Grp</mi><mo stretchy="false">(</mo><mi>G</mi><mo>,</mo><mi>H</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> (\pi_1)_{X,Y} \;\colon\; Grpd^{\ast/}((\mathbf{B}G,\ast),(\mathbf{B}H,\ast)) \longrightarrow Grp(G,H) </annotation></semantics></math></div> <p>is an equivalence.</p> <p>To see this, observe that, by definition of <a class="existingWikiWord" href="/nlab/show/pointed+objects">pointed objects</a> via the <a class="existingWikiWord" href="/nlab/show/undercategory">undercategory</a> under the point, a morphism in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Grpd</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msup></mrow><annotation encoding="application/x-tex">Grpd^{\ast/}</annotation></semantics></math> between groupoids of this form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{B}(-)</annotation></semantics></math> is a diagram in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Grp</mi></mrow><annotation encoding="application/x-tex">Grp</annotation></semantics></math> (unpointed) of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mo>*</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↙</mo></mtd> <mtd><msub><mo>⇙</mo> <mi>h</mi></msub></mtd> <mtd><mo>↘</mo></mtd></mtr> <mtr><mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mtd> <mtd></mtd> <mtd><munder><mo>⟶</mo><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>ϕ</mi></mrow></munder></mtd> <mtd></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>H</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ && \ast \\ & \swarrow &\swArrow_{h}& \searrow \\ \mathbf{B}G && \underset{\mathbf{B}\phi}{\longrightarrow} && \mathbf{B}H } </annotation></semantics></math></div> <p>where the <a class="existingWikiWord" href="/nlab/show/natural+isomorphism">natural isomorphism</a> is equivalently just the choice of an element <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi><mo>∈</mo><mi>H</mi></mrow><annotation encoding="application/x-tex">h \in H</annotation></semantics></math>. Hence these morphisms are pairs <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>ϕ</mi><mo>,</mo><mi>h</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\phi,h)</annotation></semantics></math> of a group homomorphism and an element of the domain.</p> <p>We claim that the <a class="existingWikiWord" href="/nlab/show/%282%2C1%29-functor">(2,1)-functor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">\pi_1</annotation></semantics></math> takes such <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>ϕ</mi><mo>,</mo><mi>h</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\phi,h)</annotation></semantics></math> to the homomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Ad</mi> <mrow><msup><mi>h</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup></mrow></msub><mo>∘</mo><mi>ϕ</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>G</mi><mo>⟶</mo><mi>H</mi></mrow><annotation encoding="application/x-tex">Ad_{h^{-1}} \circ \phi \;\colon\; G \longrightarrow H</annotation></semantics></math>. To see this, consider via remark <a class="maruku-ref" href="#piAs2Functor"></a> this functor as forming loops:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo>,</mo><mo>*</mo><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mo>{</mo><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mo>*</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↙</mo></mtd> <mtd></mtd> <mtd><mo>↘</mo></mtd></mtr> <mtr><mtd><mo>*</mo></mtd> <mtd></mtd> <mtd><msub><mo>⇙</mo> <mpadded width="0"><mi>g</mi></mpadded></msub></mtd> <mtd></mtd> <mtd><mo>*</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↘</mo></mtd> <mtd></mtd> <mtd><mo>↙</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mtd></mtr></mtable></mrow><mo>}</mo></mrow> <mrow><mi>g</mi><mo>∈</mo><mi>G</mi></mrow></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \pi_1(\mathbf{B}G,\ast) = \left\{ \array{ && \ast \\ & \swarrow && \searrow \\ \ast && \swArrow_{\mathrlap{g}} && \ast \\ & \searrow && \swarrow \\ && \mathbf{B}G } \right\}_{g\in G} \,. </annotation></semantics></math></div> <p>This shows that on a morphism as above this acts by forming the <a class="existingWikiWord" href="/nlab/show/pasting">pasting</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mo>*</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↙</mo></mtd> <mtd></mtd> <mtd><mo>↘</mo></mtd></mtr> <mtr><mtd><mo>*</mo></mtd> <mtd></mtd> <mtd><msub><mo>⇙</mo> <mpadded width="0"><mi>g</mi></mpadded></msub></mtd> <mtd></mtd> <mtd><mo>*</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↘</mo></mtd> <mtd></mtd> <mtd><mo>↙</mo></mtd> <mtd><msub><mo>⇙</mo> <mpadded width="0"><mi>h</mi></mpadded></msub></mtd> <mtd><mo>↘</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mtd> <mtd></mtd> <mtd><munder><mo>⟶</mo><mi>ϕ</mi></munder></mtd> <mtd></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>H</mi></mtd></mtr></mtable></mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="mediummathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mo>*</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mo>↙</mo></mtd> <mtd></mtd> <mtd><mo>↘</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo>*</mo></mtd> <mtd></mtd> <mtd><msub><mo>⇙</mo> <mpadded width="0"><mrow><mi>h</mi><mi>ϕ</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><msup><mi>h</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup></mrow></mpadded></msub></mtd> <mtd></mtd> <mtd><mo>*</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↙</mo></mtd> <mtd><msub><mo>⇙</mo> <mpadded width="0"><mi>h</mi></mpadded></msub></mtd> <mtd><mo>↘</mo></mtd> <mtd></mtd> <mtd><mo>↙</mo></mtd></mtr> <mtr><mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mtd> <mtd></mtd> <mtd><munder><mo>⟶</mo><mi>ϕ</mi></munder></mtd> <mtd></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>H</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ && \ast \\ & \swarrow && \searrow \\ \ast && \swArrow_{\mathrlap{g}} && \ast \\ & \searrow && \swarrow &\swArrow_{\mathrlap{h}}& \searrow \\ && \mathbf{B}G && \underset{\phi}{\longrightarrow} && \mathbf{B}H } \;\;\;\; = \:\;\;\; \array{ && && \ast \\ && & \swarrow && \searrow \\ && \ast && \swArrow_{\mathrlap{h\phi(g)h^{-1}}} && \ast \\ & \swarrow & \swArrow_{\mathrlap{h}} & \searrow && \swarrow \\ \mathbf{B}G && \underset{\phi}{\longrightarrow} && \mathbf{B}H } \,. </annotation></semantics></math></div> <p>Unwinding the <a class="existingWikiWord" href="/nlab/show/whiskering">whiskering</a> of <a class="existingWikiWord" href="/nlab/show/natural+transformations">natural transformations</a> here, the claim follows, as indicated by the label of the upper 2-morphisms on the right.</p> <p>One observes now that these extra labels <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi></mrow><annotation encoding="application/x-tex">h</annotation></semantics></math> are precisely the information that “trivializes” the conjugation action which in prop. <a class="maruku-ref" href="#HomsBetweenBGs"></a> prevents the bare set of group homomorphism: a <a class="existingWikiWord" href="/nlab/show/2-morphism">2-morphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>ϕ</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>h</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>⇒</mo><mo stretchy="false">(</mo><msub><mi>ϕ</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>h</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\phi_1, h_1) \Rightarrow (\phi_2,h_2)</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Grp</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msup></mrow><annotation encoding="application/x-tex">Grp^{\ast/}</annotation></semantics></math> is a natural isomorphism of groupoids</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mi>ϕ</mi> <mn>1</mn></msub></mrow></mover></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>H</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>id</mi></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd><msup><mo>⇓</mo> <mpadded width="0"><mi>h</mi></mpadded></msup></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>id</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mtd> <mtd><munder><mo>⟶</mo><mrow><msub><mi>ϕ</mi> <mn>2</mn></msub></mrow></munder></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>H</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \mathbf{B}G &\stackrel{\phi_1}{\longrightarrow}& \mathbf{B}H \\ {}^{\mathllap{id}}\downarrow &\Downarrow^{\mathrlap{h}}& \downarrow^{\mathrlap{id}} \\ \mathbf{B}G &\underset{\phi_2}{\longrightarrow}& \mathbf{B}H } </annotation></semantics></math></div> <p>(encoding a conjugation relation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ϕ</mi> <mn>2</mn></msub><mo>=</mo><msub><mi>Ad</mi> <mi>h</mi></msub><mo>∘</mo><msub><mi>ϕ</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">\phi_2 = Ad_{h} \circ \phi_1</annotation></semantics></math> as above) such that we have the <a class="existingWikiWord" href="/nlab/show/pasting">pasting</a> relation</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mo>*</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↙</mo></mtd> <mtd><msub><mo>⇙</mo> <mrow><msub><mi>h</mi> <mn>1</mn></msub></mrow></msub></mtd> <mtd><mo>↘</mo></mtd></mtr> <mtr><mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mtd> <mtd></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mi>ϕ</mi> <mn>1</mn></msub></mrow></mover></mtd> <mtd></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>H</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>id</mi></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo>⇓</mo> <mpadded width="0"><mi>h</mi></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>id</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mtd> <mtd></mtd> <mtd><munder><mo>⟶</mo><mrow><msub><mi>ϕ</mi> <mn>2</mn></msub></mrow></munder></mtd> <mtd></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>H</mi></mtd></mtr></mtable></mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mo>*</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↙</mo></mtd> <mtd><msub><mo>⇙</mo> <mrow><msub><mi>h</mi> <mn>2</mn></msub></mrow></msub></mtd> <mtd><mo>↘</mo></mtd></mtr> <mtr><mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mtd> <mtd></mtd> <mtd><munder><mo>⟶</mo><mrow><msub><mi>ϕ</mi> <mn>2</mn></msub></mrow></munder></mtd> <mtd></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>H</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ && \ast \\ & \swarrow &\swArrow_{h_1}& \searrow \\ \mathbf{B}G && \stackrel{\phi_1}{\longrightarrow} && \mathbf{B}H \\ {}^{\mathllap{id}}\downarrow && \Downarrow^{\mathrlap{h}} && \downarrow^{\mathrlap{id}} \\ \mathbf{B}G &&\underset{\phi_2}{\longrightarrow} && \mathbf{B}H } \;\;\;\;\; = \;\;\;\;\; \array{ && \ast \\ & \swarrow &\swArrow_{h_2}& \searrow \\ \mathbf{B}G && \underset{\phi_2}{\longrightarrow} && \mathbf{B}H } \,. </annotation></semantics></math></div> <p>But this says in components that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>h</mi> <mn>2</mn></msub><mo>=</mo><msub><mi>h</mi> <mn>1</mn></msub><mo>⋅</mo><mi>h</mi></mrow><annotation encoding="application/x-tex">h_2 = h_1\cdot h</annotation></semantics></math>. Hence there is a <em>at most one</em> morphism in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Grpd</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msup><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo>,</mo><mo>*</mo><mo stretchy="false">)</mo><mo>,</mo><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>H</mi><mo>,</mo><mo>*</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Grpd^{\ast/}((\mathbf{B}G,\ast),(\mathbf{B}H,\ast))</annotation></semantics></math> from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>ϕ</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>h</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\phi_1,h_1)</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>ϕ</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>h</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\phi_2,h_2)</annotation></semantics></math>: it exists if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ϕ</mi> <mn>2</mn></msub><mo>=</mo><msub><mi>Ad</mi> <mi>h</mi></msub><mo>∘</mo><msub><mi>ϕ</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">\phi_2 = Ad_h \circ \phi_1</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>h</mi> <mn>2</mn></msub><mo>=</mo><msub><mi>h</mi> <mn>1</mn></msub><mo>⋅</mo><mi>h</mi></mrow><annotation encoding="application/x-tex">h_2 = h_1\cdot h</annotation></semantics></math>.</p> <p>But since, by the previous argument, the functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">\pi_1</annotation></semantics></math> takes <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>ϕ</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>h</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\phi_1,h_1)</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Ad</mi> <mrow><msubsup><mi>h</mi> <mn>1</mn> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup></mrow></msub><mo>∘</mo><msub><mi>ϕ</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">Ad_{h_1^{-1}} \circ \phi_1</annotation></semantics></math>, this means that such a morphism exists precisely if both <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>ϕ</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>h</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\phi_1,h_1)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>ϕ</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>h</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\phi_2,h_2)</annotation></semantics></math> are taken to the same group homomorphism by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">\pi_1</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Ad</mi> <mrow><msubsup><mi>h</mi> <mn>2</mn> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup></mrow></msub><mo>∘</mo><msub><mi>ϕ</mi> <mn>2</mn></msub><mo>=</mo><msub><mi>Ad</mi> <mrow><msup><mi>h</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo>⋅</mo><msubsup><mi>h</mi> <mn>1</mn> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup></mrow></msub><mo>∘</mo><msub><mi>ϕ</mi> <mn>2</mn></msub><mo>=</mo><msub><mi>Ad</mi> <mrow><msubsup><mi>h</mi> <mn>1</mn> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup></mrow></msub><mo>∘</mo><msub><mi>ϕ</mi> <mn>1</mn></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Ad_{h_2^{-1}} \circ \phi_2 = Ad_{h^{-1}\cdot h_1^{-1}}\circ \phi_2 = Ad_{h_1^{-1}} \circ \phi_1 \,. </annotation></semantics></math></div> <p>This establishes that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">\pi_1</annotation></semantics></math> is also an equivalence on all <a class="existingWikiWord" href="/nlab/show/hom-groupoids">hom-groupoids</a>.</p> </div> <p>This proof also shows that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{B}(-)</annotation></semantics></math> is in fact the inverse equivalence:</p> <div class="num_cor"> <h6 id="corollary">Corollary</h6> <p>There is an <a class="existingWikiWord" href="/nlab/show/equivalence+of+%282%2C1%29-categories">equivalence of (2,1)-categories</a> between pointed connected groupoids and plain groups</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Grp</mi><mover><munder><mo>⟶</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle></munder><munderover><mo>⟵</mo><mo>≃</mo><mrow><msub><mi>π</mi> <mn>1</mn></msub><mo>=</mo><msub><mi>Ω</mi> <mo>*</mo></msub></mrow></munderover></mover><msubsup><mi>Grpd</mi> <mrow><mo>≥</mo><mn>1</mn></mrow> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msubsup></mrow><annotation encoding="application/x-tex"> Grp \stackrel{\underoverset{\simeq}{\pi_1 = \Omega_\ast}{\longleftarrow}}{\underset{\mathbf{B}}{\longrightarrow}} Grpd^{\ast/}_{\geq 1} </annotation></semantics></math></div> <p>given by forming <a class="existingWikiWord" href="/nlab/show/loop+space+objects">loop space objects</a> and by forming deloopings.</p> </div> <h3 id="for_topological_spaces_and_groupoids">For topological spaces and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-groupoids</h3> <p>There is an <a class="existingWikiWord" href="/nlab/show/equivalence+of+%28%E2%88%9E%2C1%29-categories">equivalence of (∞,1)-categories</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mn>∞</mn><msubsup><mi>Grpd</mi> <mrow><mo>≥</mo><mn>1</mn></mrow> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msubsup><mover><munder><mo>→</mo><mi>Ω</mi></munder><mover><mo>←</mo><mi>B</mi></mover></mover><mn>∞</mn><mi>Group</mi></mrow><annotation encoding="application/x-tex"> \infty Grpd^{\ast/}_{\geq 1} \stackrel{\overset{B}{\leftarrow}}{\underset{\Omega}{\to}} \infty Group </annotation></semantics></math></div> <p>between <a class="existingWikiWord" href="/nlab/show/pointed+object">pointed</a> <a class="existingWikiWord" href="/nlab/show/connected">connected</a> <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groupoid">∞-groupoid</a>s and <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group">∞-group</a>s, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ω</mi></mrow><annotation encoding="application/x-tex">\Omega</annotation></semantics></math> forms <a class="existingWikiWord" href="/nlab/show/loop+space+object">loop space object</a>s.</p> <p>This is <a class="existingWikiWord" href="/nlab/show/presentable+%28%E2%88%9E%2C1%29-category">presented</a> by a <a class="existingWikiWord" href="/nlab/show/Quillen+equivalence">Quillen equivalence</a> of <a class="existingWikiWord" href="/nlab/show/model+categories">model categories</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>sSet</mi> <mo>*</mo></msub><mover><munder><mo>→</mo><mi>G</mi></munder><mover><mo>←</mo><mover><mi>W</mi><mo stretchy="false">¯</mo></mover></mover></mover><mi>sGrp</mi></mrow><annotation encoding="application/x-tex"> sSet_* \stackrel{\overset{\bar W}{\leftarrow}}{\underset{G}{\to}} sGrp </annotation></semantics></math></div> <p>between the <a class="existingWikiWord" href="/nlab/show/model+structure+on+reduced+simplicial+sets">model structure on reduced simplicial sets</a> and the <a class="existingWikiWord" href="/nlab/show/transferred+model+structure">transferred model structure</a> on <a class="existingWikiWord" href="/nlab/show/simplicial+group">simplicial group</a>s along the <a class="existingWikiWord" href="/nlab/show/forgetful+functor">forgetful functor</a> to the <a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+sets">model structure on simplicial sets</a>.</p> <p>(See <a class="existingWikiWord" href="/nlab/show/groupoid+object+in+an+%28infinity%2C1%29-category">groupoid object in an (infinity,1)-category</a> for more details on this Quillen equivalence.)</p> <h3 id="for_parameterized_groupoids_stacks__sheaves">For parameterized <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-groupoids (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-stacks / <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-sheaves)</h3> <p>The following result makes precise for <em>parameterized <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groupoid">∞-groupoid</a>s</em> – for <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-stack">∞-stack</a>s – the general statement that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>-fold <a class="existingWikiWord" href="/nlab/show/delooping">delooping</a> provides a correspondence between <a class="existingWikiWord" href="/nlab/show/n-category">n-categories</a> that have trivial <a class="existingWikiWord" href="/nlab/show/k-morphism">r-morphism</a>s for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>r</mi><mo><</mo><mi>k</mi></mrow><annotation encoding="application/x-tex">r \lt k</annotation></semantics></math> and <a class="existingWikiWord" href="/nlab/show/k-tuply+monoidal+n-category">k-tuply monoidal n-categories</a>.</p> <div class="num_defn"> <h6 id="definition">Definition</h6> <p>An <a class="existingWikiWord" href="/nlab/show/Ek-algebra">Ek-algebra</a> object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> in an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math> is called <strong>groupal</strong> if its <a class="existingWikiWord" href="/nlab/show/homotopy+groups+in+an+%28%E2%88%9E%2C1%29-topos">connected components</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>∈</mo><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><mo>≤</mo><mn>0</mn></mrow></msub></mrow><annotation encoding="application/x-tex">\pi_0(A) \in \mathbf{H}_{\leq 0}</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/group+object">group object</a>.</p> <p>Write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>Mon</mi> <mrow><mi>𝔼</mi><mo stretchy="false">[</mo><mi>k</mi><mo stretchy="false">]</mo></mrow> <mi>gp</mi></msubsup><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Mon^{gp}_{\mathbb{E}[k]}(\mathbf{H})</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a> of groupal <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">E_k</annotation></semantics></math>-algebra objects in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math>.</p> <p>A groupal <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">E_1</annotation></semantics></math>-algebra – hence an groupal <a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E+algebra">A-∞ algebra</a> object in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math> – we call an <em><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group">∞-group</a></em> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math>. Write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn><mi>Grp</mi><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\infty Grp(\mathbf{H})</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a> of <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group">∞-group</a>s in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math>.</p> </div> <div class="num_theorem" id="LoopingDeloopingForHigherStacks"> <h6 id="theorem">Theorem</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>></mo><mn>0</mn></mrow><annotation encoding="application/x-tex">k \gt 0</annotation></semantics></math>, let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math> be an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+of+%28%E2%88%9E%2C1%29-sheaves">(∞,1)-category of (∞,1)-sheaves</a> and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mo>*</mo> <mrow><mo>≥</mo><mi>k</mi></mrow></msubsup></mrow><annotation encoding="application/x-tex">\mathbf{H}_*^{\geq k}</annotation></semantics></math> denote the <a class="existingWikiWord" href="/nlab/show/full+subcategory">full subcategory</a> of the category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mo>*</mo></msub></mrow><annotation encoding="application/x-tex">\mathbf{H}_{*}</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/pointed+objects">pointed objects</a>, spanned by those pointed objects that are <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">k-1</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/connected">connected</a> (i.e. their first <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/homotopy+groups+in+an+%28%E2%88%9E%2C1%29-topos">homotopy sheaves</a>) vanish. Then there is a canonical equivalence of <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-categories</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><mo>≥</mo><mi>k</mi></mrow> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msubsup><mo>≃</mo><msubsup><mi>Mon</mi> <mrow><mi>𝔼</mi><mo stretchy="false">[</mo><mi>k</mi><mo stretchy="false">]</mo></mrow> <mi>gp</mi></msubsup><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbf{H}^{\ast/}_{\geq k} \simeq Mon^{gp}_{\mathbb{E}[k]}(\mathbf{H}) \,. </annotation></semantics></math></div> <p>between the pointed <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>k</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(k-1)</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/connected">connected</a> objects and the groupal <a class="existingWikiWord" href="/nlab/show/Ek-algebra">Ek-algebra</a> objects in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math>.</p> </div> <p>This is <em><a class="existingWikiWord" href="/nlab/show/Higher+Algebra">Higher Algebra, theorem 5.2.6.10</a></em></p> <p>Specifically for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo>=</mo></mrow><annotation encoding="application/x-tex">\mathbf{H} = </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Top">Top</a>, this reduces to the following classical theorem due to <a class="existingWikiWord" href="/nlab/show/Peter+May">Peter May</a>, known as the <a class="existingWikiWord" href="/nlab/show/May+recognition+theorem">May recognition theorem</a>.</p> <div class="num_theorem"> <h6 id="theorem_2">Theorem</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a> equipped with an action of the <a class="existingWikiWord" href="/nlab/show/little+cubes+operad">little cubes operad</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒞</mi> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">\mathcal{C}_k</annotation></semantics></math> and suppose that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> is grouplike. Then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> is homotopy equivalent to a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>-fold loop space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Ω</mi> <mi>k</mi></msup><mi>X</mi></mrow><annotation encoding="application/x-tex">\Omega^k X</annotation></semantics></math> for some pointed topological space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>.</p> </div> <p>This is <em><a class="existingWikiWord" href="/nlab/show/Higher+Algebra">Higher Algebra, theorem 5.2.6.15</a></em></p> <div class="num_remark" id="LoopingDeloopingDegree1InTopos"> <h6 id="remark_2">Remark</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">k = 1</annotation></semantics></math> we have a looping/delooping equivalence</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Grp</mi><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">)</mo><mover><munder><mo>⟶</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle></munder><mover><mo>⟵</mo><mi>Ω</mi></mover></mover><msubsup><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><mo>≥</mo><mn>1</mn></mrow> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msubsup></mrow><annotation encoding="application/x-tex"> Grp(\mathbf{H}) \stackrel{\overset{\Omega}{\longleftarrow}}{\underset{\mathbf{B}}{\longrightarrow}} \mathbf{H}_{\geq 1}^{\ast /} </annotation></semantics></math></div> <p>between pointed connected objects in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math> and grouplike <a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E+algebra">A-∞ algebra</a> objects in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math>: <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group">∞-group</a> objects in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math>.</p> </div> <div class="num_remark"> <h6 id="remark_3">Remark</h6> <p>If the ambient <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a> has <a class="existingWikiWord" href="/nlab/show/homotopy+dimension">homotopy dimension</a> 0 then every connected object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math> admits a point <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>*</mo><mo>→</mo><mi>E</mi></mrow><annotation encoding="application/x-tex">* \to E</annotation></semantics></math>. Still, the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a> of pointed connected objects differs from that of unpointed connected objects (because in the latter the <a class="existingWikiWord" href="/nlab/show/natural+transformation">natural transformation</a>s may have nontrivial components on the point, while in the former case they may not).</p> <p>The connected objects <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math> which fail to be <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group">∞-group</a>s by failing to admit a point are of interest: these are the <em><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-gerbes">∞-gerbes</a></em> in the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a>.</p> </div> <h3 id="for_cohesive_groupoids">For cohesive <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-groupoids</h3> <p>A special case of the parameterized <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-groupoids above are <a class="existingWikiWord" href="/nlab/show/cohesive+%E2%88%9E-groupoid">cohesive ∞-groupoid</a>s. Looping and delooping for these is discussed at <a class="existingWikiWord" href="/nlab/show/cohesive+%28%E2%88%9E%2C1%29-topos+--+structures">cohesive (∞,1)-topos – structures</a> in the section <a href="http://www.ncatlab.org/nlab/show/cohesive%20%28infinity,1%29-topos%20--%20structures#InfinGroups">Cohesive ∞-groups</a>.</p> <h3 id="for_categories">For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,n)</annotation></semantics></math>-categories</h3> <p>See <a class="existingWikiWord" href="/nlab/show/delooping+hypothesis">delooping hypothesis</a>.</p> <h2 id="relation_to_looping_and_suspension">Relation to looping and suspension</h2> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> any monoidal <a class="existingWikiWord" href="/nlab/show/space">space</a>, we may <a class="existingWikiWord" href="/nlab/show/forgetful+functor">forget</a> its monoidal structure and just remember the underlying <a class="existingWikiWord" href="/nlab/show/space">space</a>. The formation of <a class="existingWikiWord" href="/nlab/show/loop+space+object">loop space object</a>s composed with this <a class="existingWikiWord" href="/nlab/show/forgetful+functor">forgetful functor</a> has a <a class="existingWikiWord" href="/nlab/show/left+adjoint">left adjoint</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math> which forms <em><a class="existingWikiWord" href="/nlab/show/suspension+objects">suspension objects</a></em>.</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/looping">looping</a>, <a class="existingWikiWord" href="/nlab/show/delooping">delooping</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/May+recognition+theorem">May recognition theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Quillen+equivalence+between+simplicial+groups+and+reduced+simplicial+sets">Quillen equivalence between simplicial groups and reduced simplicial sets</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pointed+connected+groupoid">pointed connected groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/loop+space+object">loop space object</a>, <a class="existingWikiWord" href="/nlab/show/free+loop+space+object">free loop space object</a>,</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/delooping">delooping</a>, <strong>looping and delooping</strong></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/loop+space">loop space</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/free+loop+space">free loop space</a>, <a class="existingWikiWord" href="/nlab/show/derived+loop+space">derived loop space</a>, <a class="existingWikiWord" href="/nlab/show/free+loop+stack">free loop stack</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/suspension+object">suspension object</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/suspension">suspension</a>, <a class="existingWikiWord" href="/nlab/show/reduced+suspension">reduced suspension</a></li> </ul> </li> </ul> <div> <table><thead><tr><th></th><th><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-operad">(∞,1)-operad</a></th><th><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-algebra+over+an+%28%E2%88%9E%2C1%29-operad">∞-algebra</a></th><th>grouplike version</th><th>in <a class="existingWikiWord" href="/nlab/show/Top">Top</a></th><th>generally</th></tr></thead><tbody><tr><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E+operad">A-∞ operad</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E+algebra">A-∞ algebra</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group">∞-group</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E+space">A-∞ space</a>, e.g. <a class="existingWikiWord" href="/nlab/show/loop+space">loop space</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/loop+space+object">loop space object</a></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/E-k+operad">E-k operad</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/E-k+algebra">E-k algebra</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/k-monoidal+%E2%88%9E-group">k-monoidal ∞-group</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/iterated+loop+space">iterated loop space</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/iterated+loop+space+object">iterated loop space object</a></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/E-%E2%88%9E+operad">E-∞ operad</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/E-%E2%88%9E+algebra">E-∞ algebra</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/abelian+%E2%88%9E-group">abelian ∞-group</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/E-%E2%88%9E+space">E-∞ space</a>, if grouplike: <a class="existingWikiWord" href="/nlab/show/infinite+loop+space">infinite loop space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>≃</mo></mrow><annotation encoding="application/x-tex">\simeq</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-space">∞-space</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/infinite+loop+space+object">infinite loop space object</a></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>≃</mo></mrow><annotation encoding="application/x-tex">\simeq</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/connective+spectrum">connective spectrum</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>≃</mo></mrow><annotation encoding="application/x-tex">\simeq</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/connective+spectrum+object">connective spectrum object</a></td></tr> <tr><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/stabilization">stabilization</a></strong></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/spectrum">spectrum</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/spectrum+object">spectrum object</a></td></tr> </tbody></table> <ul> <li><a class="existingWikiWord" href="/nlab/show/looping+and+delooping">looping and delooping</a>, <a class="existingWikiWord" href="/nlab/show/stabilization+hypothesis">stabilization hypothesis</a></li> </ul> </div> <h2 id="references">References</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Jacob+Lurie">Jacob Lurie</a>, Section 6.1.2 of: <em><a class="existingWikiWord" href="/nlab/show/Higher+Topos+Theory">Higher Topos Theory</a></em></p> </li> <li id="LurieAlgebra"> <p><a class="existingWikiWord" href="/nlab/show/Jacob+Lurie">Jacob Lurie</a>,</p> <p>Section 5.1.3 of: <em><a class="existingWikiWord" href="/nlab/show/Higher+Algebra">Higher Algebra</a></em></p> </li> </ul> <p>Discussion in <a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theory</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/David+W%C3%A4rn">David Wärn</a>, <em>Eilenberg-MacLane spaces and stabilisation in homotopy type theory</em> [<a href="https://arxiv.org/abs/2301.03685">arXiv:2301.03685</a>]</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on November 7, 2024 at 00:21:26. See the <a href="/nlab/history/looping" style="color: #005c19">history</a> of this page for a list of all contributions to it. </p> </div> <div class="navigation navfoot"> <a href="/nlab/edit/looping" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussion/11586/#Item_2">Discuss</a><span class="backintime"><a href="/nlab/revision/looping/27" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a></span><a href="/nlab/show/diff/looping" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Changes from previous revision</a><a href="/nlab/history/looping" accesskey="S" class="navlink" id="history" rel="nofollow">History (27 revisions)</a> <a href="/nlab/show/looping/cite" style="color: black">Cite</a> <a href="/nlab/print/looping" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/looping" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>